Solve the inequation `5x - 5 le - 5` If (i) x `in` z (ii) `x `in R and represent the solution on the number line

Explanation

To solve the inequation \(5x - 5 \leq -5\), we will follow these steps:

### Step 1: Write down the inequation
The given inequation is:
\[ 5x - 5 \leq -5 \]

### Step 2: Add 5 to both sides
To isolate the term with \(x\), we add 5 to both sides of the inequation:
\[ 5x - 5 + 5 \leq -5 + 5 \]
This simplifies to:
\[ 5x \leq 0 \]

### Step 3: Divide both sides by 5
Next, we divide both sides by 5 to solve for \(x\):
\[ \frac{5x}{5} \leq \frac{0}{5} \]
This simplifies to:
\[ x \leq 0 \]

### Step 4: Analyze the solution for \(x \in \mathbb{Z}\)
Now we will consider the case when \(x\) belongs to the set of integers (\(\mathbb{Z}\)):
- The integers that satisfy \(x \leq 0\) are:
\[ \{..., -3, -2, -1, 0\} \]

### Step 5: Represent the solution on the number line for \(x \in \mathbb{Z}\)
On the number line, we will represent the integers less than or equal to 0:
- Mark the point 0 and shade all points to the left (towards negative infinity).

### Step 6: Analyze the solution for \(x \in \mathbb{R}\)
Now we will consider the case when \(x\) belongs to the set of real numbers (\(\mathbb{R}\)):
- The solution \(x \leq 0\) includes all real numbers less than or equal to 0.

### Step 7: Represent the solution on the number line for \(x \in \mathbb{R}\)
On the number line, we will represent all real numbers less than or equal to 0:
- Mark the point 0 and shade all points to the left (towards negative infinity), including 0.

### Final Summary of Solutions:
1. For \(x \in \mathbb{Z}\): The solution is \(\{..., -3, -2, -1, 0\}\).
2. For \(x \in \mathbb{R}\): The solution is all real numbers \(x \leq 0\).

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